Cutting the face it create extra geometry
its because the boolean operator works only on volumetric meshes.
say tube and cube, but not tube and plane. a plane is flat.

also sometimes it goes wrong because you think the shape is 3d but its not some flaw, resulting in a non watertight shape.

in your case you might apply a solidify modifier to the plane and then go on with what you did. (you need to apply it, not only add the modifier).

happy blending.